We also observe that since \(\tau \)  is realizable, \(\mathfrak {M}(\mathcal {X},\tau )\)  is reduced by Proposition \(3.28\)  of [1]}, and so is \(\mathfrak {M}^{ev}(\mathcal {X},\tau )\) . Thus, the morphism \(\mathfrak {M}^{ev}(\mathcal {X},\tau )_s \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\tau )\)  induces a morphism \(\mathfrak {M}^{ev}(\mathcal {X},\tau )_{s,red} \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\tau )\) . To relate the invariant of interest to an integral against \([\mathcal {M}(X,\tau )]^{vir}\) , we need to calculate the degree of \(\mathfrak {M}^{ev}(\mathcal {X},\tau )_{s,red} \rightarrow \mathfrak {M}^{ev}(\mathcal {X},\tau )\) . This calculation is as in [2]} Section 6 step 4, and is given by \(|coker(P_\mathfrak {u}^{gp} \rightarrow Q^{gp} \oplus \mathbb {Z}^n)_{tor}|\) . Furthermore, by definition of the fine and saturated fiber product, we have \(Q_\xi ^{gp} = coker(P_\mathfrak {u}^{gp} \rightarrow Q_{\tau }^{gp} \oplus \mathbb {Z}^n)/tor\) . Now consider the following morphisms of exact sequences of abelian groups:
\(\begin{tikzcd}0{r} & P_{\mathfrak {u}}^{gp} {r} {d} & P_{\mathfrak {u}}^{gp} \oplus \mathbb {Z}^n {r} {d} & \mathbb {Z}^n {r} {d} & 0 \\0 {r} & Q_\tau ^{gp} \oplus \mathbb {Z}^n {r} & Q_\tau ^{gp} \oplus \mathbb {Z}^n {r} & 0 {r} & 0\end{tikzcd}\) 
